The stock calendar app can sync with your Gmail calendar but you need to make sure you select your Google calendar each time you create an event instead of "My Calendar". There's no way to make that the default automatically. Other 3rd-party calendar apps can... check out Jorte and Gemini.

Unfortunately Google has yet to release an API for tasks so there's no easy way to get a tasks app to sync with your Google account. I use Astrid Tasks which has its own syncing service.

Same with notes... I use AK Notepad, which has its own online sync service. Sucks to have 3 separate online places to sync things too, but yell at Google for not being as open as they pretend to be. Until they release the APIs there's not much that can be done.